Partial differential equations : an introduction to analytical and numerical methods / Wolfgang Arendt, Karsten Urban ; translated from the German by James B. Kennedy.
2022
QA374 .A74 2022
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Title
Partial differential equations : an introduction to analytical and numerical methods / Wolfgang Arendt, Karsten Urban ; translated from the German by James B. Kennedy.
Uniform Title
Partielle Differenzialgleichungen. English
ISBN
9783031133794 electronic book
303113379X electronic book
9783031133787
3031133781
303113379X electronic book
9783031133787
3031133781
Published
Cham : Springer, 2022.
Language
English
Description
1 online resource (1 volume) : illustrations (black and white)
Item Number
10.1007/978-3-031-13379-4 doi
Call Number
QA374 .A74 2022
Dewey Decimal Classification
515/.353
Summary
This textbook introduces the study of partial differential equations using both analytical and numerical methods. By intertwining the two complementary approaches, the authors create an ideal foundation for further study. Motivating examples from the physical sciences, engineering, and economics complete this integrated approach. A showcase of models begins the book, demonstrating how PDEs arise in practical problems that involve heat, vibration, fluid flow, and financial markets. Several important characterizing properties are used to classify mathematical similarities, then elementary methods are used to solve examples of hyperbolic, elliptic, and parabolic equations. From here, an accessible introduction to Hilbert spaces and the spectral theorem lay the foundation for advanced methods. Sobolev spaces are presented first in dimension one, before being extended to arbitrary dimension for the study of elliptic equations. An extensive chapter on numerical methods focuses on finite difference and finite element methods. Computer-aided calculation with Maple completes the book. Throughout, three fundamental examples are studied with different tools: Poissons equation, the heat equation, and the wave equation on Euclidean domains. The BlackScholes equation from mathematical finance is one of several opportunities for extension. Partial Differential Equations offers an innovative introduction for students new to the area. Analytical and numerical tools combine with modeling to form a versatile toolbox for further study in pure or applied mathematics. Illuminating illustrations and engaging exercises accompany the text throughout. Courses in real analysis and linear algebra at the upper-undergraduate level are assumed.
Note
Translated from the German.
Bibliography, etc. Note
Includes bibliographical references.
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Access limited to authorized users.
Source of Description
Description based on print version record.
Added Author
Urban, Karsten, author.
Kennedy, J. B., translator.
Kennedy, J. B., translator.
Series
Graduate texts in mathematics ; 294.
Available in Other Form
Partial differential equations.
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Table of Contents
1 Modeling, or where do differential equations come from
2 Classification and characteristics
3 Elementary methods
4 Hilbert spaces
5 Sobolev spaces and boundary value problems in dimension one
6 Hilbert space methods for elliptic equations
7 Neumann and Robin boundary conditions
8 Spectral decomposition and evolution equations
9 Numerical methods
10 Maple, or why computers can sometimes help
Appendix.
2 Classification and characteristics
3 Elementary methods
4 Hilbert spaces
5 Sobolev spaces and boundary value problems in dimension one
6 Hilbert space methods for elliptic equations
7 Neumann and Robin boundary conditions
8 Spectral decomposition and evolution equations
9 Numerical methods
10 Maple, or why computers can sometimes help
Appendix.